## Monday, June 23, 2014

### June 2014 Geometry Regents, Part 1

I just got home from grading the Geometry Regents exams in Staten Island, so I finally had a good look at the exam. Students and teachers alike had both described it as a "fair" exam (sort of like the Algebra test), and I'd have to agree.

Here are the multiple-choice questions from Part I. Keep in mind, that I have to type all of these, so the rest of the test may not show up on my blog as quickly as you may like. Questions are always welcome. Likewise, because I've been asked to hurry with this, there are no diagrams included. They may get added at a later time.

1. Plane P is parallel to plane Q. If plane P is perpendicular to line l, then plane Q...

(3) is perpendicular to line l. Think of a fire pole at a fire station going through the second floor to the first floor.

2. In the diagram below [Diagram Omitted], quadrilateral ABCD has vertices A(-5,1), B(6, -1), C(3, 5) and D(-2, 7). What are the coordinates of the midpoint of diagonal AC?

The midpoint is the average of the two x-values and the average of the two y-values. ( (-5+3)/2, (1+5)/2 ), which is (-1, 3).

3. In the diagram below, transversal TU intersects PQ and RS at V and W, respectively. If m<TVQ = 5x - 22, and m<VWS = 3x + 10, for which value of x is PQ || RS?

You want the value of x that makes 5x - 22 = 3x + 10, because corresponding angles are congruent when parallel lines are cut by a transversal. Solve for x and you get 16.

4. The measures of the angels of a triangle are in the ratio 2:3:4. In degrees, the measure of the largest angle of the triangle is:

Take the ratio and write the following equation: 2x + 3x + 4x = 180. So 9x = 180, and x = 20. The largest angle is 4x, which is 4(20) = 80 degrees.

5. The diamter of the base of a right circular cylinder is 6 cm and the height is 15 cm. In square centimeters, the lateral area of the cylinder is

The lateral area is the circumference of the base times the height, or (pi)(d)(h), which is (pi)(6)(15), which is 90*pi.

6. When the system of equations y + 2x = x2 and y = x is graphed on a set of axes, what is the total number of points of intersection?

Substitute y = x into the other equation and you get x + 2x = x2
This can be rewritten as x2 - 3x = 0.
You don't need to solve it to know that there are two distinct solutions. You could have graphed the equations as well. Note: A parabola and a straight line can have 0, 1 or 2 intersection points. They cannot have three.

7. The vertex angle of an isosceles triangle measures 15 degrees more than one of its base angles. How many degrees are there in a base angle of the triangle?

The sum of the angles is x + x + x + 15 = 180. So 3x + 15 = 180, 3x = 165 and x = 55, which is the measure of one base angle.

8. Circle O is graphed on the set of axes below. [Diagram omitted] Which equation represents circle O?

The correct form is (x - h)2 + (y - k)2 = r2, where (h, k) is the center of the circle and r is the radius.
In the graph, the center is (-1, 3) and the radius is 3.
So the equation is (x + 1)2 + (y - 3)2 = 9.

9. In the diagram of the circle shown below [Diagram omitted], chords AC and BD intersect at Q and chords AE and BD are parallel. Which statement must always be true?

Arcs AB and DE are congruent because the arcs formed by two parallel chords are always congruent.

10. In the diagram below [Diagram omitted], triangle AEC is congruent to triangle BED. Which statement is not always true?

Angle ACE does not correspond to angle DBE, so they will not always be congruent although they could be in an isosceles or equilateral triangle.

11. What is the length of RS with R(-2, 3) and S(4, 5)?

Distance formula, a.k.a. Pythagorean Theorem. The square root of (62 + 22 is the square root of 40, which simplifies to 2 times square root of 10.

12. What are the truth values of the statement "Two is prime" and its negation?

Two is a prime number so the statement is True, which means the negation is False. Choices 2 and 3 are just silly and should have been eliminated immediately.

13. A regular polygon has an exterior angle that measures 45 degrees. How many sides does the polygon have?

If you didn't know just from constant repetition of the problem in class, remember that 360/n = 45, so n = 360/45 = 8 sides.

14. In rhombus ABCD with diagonals AC and DB, AD = 10. if the length of diagonal AC is 12, what is the length of DB?

If one side is 10, all sides are 10. The diagonals are perpendicular bisectors. This means that there are four little right triangles with hypotenuse 10. One leg of each triangle is 6 (12/2). Using Pythagorean Theorem, or just knowing the triple, the other leg of each triangle is 8. That makes the length of the entire diagonal DB 16.

15. If the surface area of a sphere is 144*pi square centimeters, what is the length of the diameter of the sphere, in centimeters?

The surface area of a sphere (given in the back of the booklet) is 4*pi*r2 = 144 * pi. Divide both sides by 4*pi, and r2 = 36, so r = 6.
The radius is half the diameter, which is 12.

16. Which numbers could represent the lengths of a the sides of a triangle?

Not a right triangle, just a triangle. The rule for the sides of a triangle is that the sum of the two smaller sides must be more than the length of the largest side. 5 + 9 = 14, 7 + 7 < 15, 1 + 2 < 4, 3 + 6 > 8.

17. The equation of a line is 3y + 2x = 12. What is the slope of the line perpendicular to the given line?

The slopes of two perpendicular lines are inverse reciprocals. (That is, change the sign, and flip the fraction over.)
First you have to find the slope of the given line. Subtract 2x from both sides and divide everything by 3 (the coefficient of y). The slope of the given line is -2/3.
That makes the perpendicular slope 3 / 2. CORRECTION.

18. In the diagram below, point K is in plane P. How many lines can be drawn through K, perpendicular to plane P?

Similar to the other question, only one line perpendicular to a plane can be drawn at any one point. Think of a pencil through a hole in your loose leaf.

19. In the diagram below, AB and CD are bases of trapezoid ABCD. (Not drawn to scale) If m<B = 123 and m<D = 73, what is the m<C?

Because it isn't an isosceles trapezoid, <C and <D can not be assumed to be congruent. However, <B and <C are supplementary. So the answer is 57.

20. What is the equation of a line passing through the point (4, -1) and parallel to the line whose equation is 2y - x = 8?

Rewrite 2y - x = 8 as y = (1/2) x + 4, slope is 1/2. So the parallel line must have the equation y = (1/2) x + b.
Plug is 4 for x and -1 for y and solve for b. You will find that b = -3, so the answer is y = (1/2) x - 3.

21. The image of rhombus VWXY preserves which properties under the transformation T2, -3?

Translations do not change shape nor orientation, so the correct answer is both parallelism and orientation.

22. The equation of a circle is (x - 3)2 + y2 = 8. The coordinates of its center and the length of its radius are

Using the form given above, the center is (3, 0) and the radius is the square root of 8, which is 2 times the square root of 2.

23. Which statement has the same truth value as the statement, "If a quadrilateral is a square, then it is a rectangle"?

Note that it didn't ask what the truth value of the statement was. That isn't important. What is important is that the statement has the same truth value as its contrapositive, which is "If a quadrilateral is not a rectangle, then it is not a square."

24. The three medians of a triangle intersect at a point. Which measurements could represent the segments of one of the medians?

The centroid occurs 2/3rds the way down the length of the median. In other words, the two segments will have a ratio of 1:2, with each segment being 1/3 or 2/3 the length of the median itself. The correct choice is 3 and 6.

25. In the diagram of triangle PQR shown below {NO, IT ISN'T], PR is extended to S, m<P = 110, m<Q = 4x and m<QRS = x2 + 5x. What is m<Q?

The sum of the measures of angles P and Q is equal to the exterior angle R. The equation to write is x2 + 5x = 4x + 110.
This becomes a quadratic equation, x2 + x - 110 = 0.
This factors in (x + 11)(x - 10) = 0, so x = -11 (discard the negative) or x = 10.
HOWEVER, they want the size of the angle, not of x. The size of the angle is 4x = 4(10) = 40.

26. Triangle PQT with RS || QT is shown below [DONT HOLD YOUR BREATH]. If PR = 12, RQ = 8 and PS = 21, what is the length of PT?

You can set up a proportion (21 / x) = (12 / 8), which will find the length of ST, which is 14. PT = 21 + 14, which is 35.

27. In the diagram of line segment WXYZ below, WY is congruent to XZ. Which reasons can be used to prove VW is congruent to YZ?

The reflexive property (XY is congruent to XY) and subtraction postulate (WY - XY is congruent to XZ - XY).

28. The coordinates of the endpoints of the diamter of a circle are (2, 0) and (2, -8). What is the equation of the circle?

This is the third question about the equation of a circle. The center of the circle is (2, -4). The radius is 4.
The equation of the circle is (x - 2)2 + (y + 4)2 = 16.

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So how did everyone do? More importantly, how did I do? I mean, I did rush this. Mistakes happen.